Loop Order Research Articles

Algorithms for minimal Picard–Fuchs operators of Feynman integrals

In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function in projective space. By using an algorithm that extends the Griffiths--Dwork reduction for the case of projective hypersurfaces with singularities, we derive Fuchsian linear differential equations, the Picard--Fuchs equations, with respect to kinematic parameters for a large class of massive multi-loop Feynman integrals. With this approach we obtain the differential operator for Feynman integrals to high multiplicities and high loop orders. Using recent factorisation algorithms we give the minimal order differential operator in most of the cases studied in this paper. Amongst our results are that the order of Picard--Fuchs operator for the generic massive two-point $n-1$-loop sunset integral in two-dimensions is $2^{n}-\binom{n+1}{\left\lfloor \frac{n+1}{2}\right\rfloor }$ supporting the conjecture that the sunset Feynman integrals are relative periods of Calabi--Yau of dimensions $n-2$. We have checked this explicitly till six loops. As well, we obtain a particular Picard--Fuchs operator of order 11 for the massive five-point tardigrade non-planar two-loop integral in four dimensions for generic mass and kinematic configurations, suggesting that it arises from $K3$ surface with Picard number 11. We determine as well Picard--Fuchs operators of two-loop graphs with various multiplicities in four dimensions, finding Fuchsian differential operators with either Liouvillian or elliptic solutions.

CNNFlow: Memory-driven Data Flow Optimization for Convolutional Neural Networks

Convolution Neural Networks (CNNs) are widely deployed in computer vision applications. The datasets are large, and the data reuse across different parts is heavily interleaved. Given that memory access (SRAM and especially DRAM) is more expensive in both performance and energy than computation, maximizing data reuse to reduce data movement across the memory hierarchy is critical to improving execution efficiency. This is even more important for the common use case of CNNs on mobile devices where computing/memory resources are limited. We propose CNNFlow, a memory-driven dataflow optimization framework to automatically schedule CNN computation on a given CNN architecture to maximize data reuse at each level of the memory hierarchy. We provide a mathematical calculation for data reuses in terms of parameters including loop ordering, blocking, and memory-bank allocation for tensors in CNN. We then present a series of techniques that help prune the large search space and reduce the cost of the exploration. This provides, for the first time, an exact and practical search algorithm for optimal solutions to minimize memory access cost for CNN. The efficacy is demonstrated for two widely used CNN algorithms: AlexNet and VGG16 with 5 and 13 convolution layers, respectively. CNNFlow finds the optimal solution for each layer within tens of minutes of compute time. Its solution requires about 20% fewer DRAM accesses and 40%–80% fewer SRAM accesses compared to state-of-the-art algorithms in the literature.

Exact Off Shell Sudakov Form Factor in N=4 Supersymmetric Yang-Mills Theory.

We consider the Sudakov form factor in planar N=4 supersymmetric Yang-Mills theory in the off shell kinematical regime, which can be achieved by considering the theory on its Coulomb branch. We demonstrate that for up to three loops both the infrared-divergent as well as the finite terms do exponentiate, with the coefficient accompanying log^{2}(m^{2}) determined by the octagon anomalous dimension Γ_{oct}. This behavior is in stark contrast to previous conjectural accounts in the literature. Together with the finite terms we observe that for up to three loops the logarithm of the Sudakov form factor is identical to twice the logarithm of the null octagon O_{0}, which was recently introduced within the context of integrability-based approaches to four point correlation functions with infinitely large R charges. The null octagon O_{0} is known in a closed form for all values of the 't Hooft coupling constant and kinematical parameters. We conjecture that the relation between O_{0} and the off shell Sudakov form factor will hold to all loop orders.

Surface Stabilizing Speed-Tracking Control for PMSMs via Loop Adaptation and Order Reduction Approaches

The proposed single-loop solution to the permanent-magnet synchronous motor (PMSM) speed-tracking problem provides improved transient performance and robustness by overcoming the two challenging points of parameter variation and performance inconsistency due to load changes. The features are summarized as follows: 1) the proportional-type control law exponentially stabilizes the desired surface with the disturbance observers (DOBs) through the order reduction property, resulting in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -axis current independent single-loop structure for speed control with first-order speed error dynamics; 2) the parameter-independent angular acceleration observer renders the proportional-type surface-stabilizing control implementable without an additional sensor and direct time derivative operation based on the speed measurement; and 3) the adaptive convergence rate mechanism in the analytic form provides improved transient performance through the loop adaptation of the closed-loop system without requiring an additional optimization process. The experimental study demonstrates the practical advantages using a 1-kW PMSM drive system.

Step towards a consistent treatment of chiral theories at higher loop order: The abelian case

As recently pointed out, regularization schemes defined in four-dimensions may also face inconsistencies in the presence of chiral fermions. In this work, we extend this analysis to two-loop order. Adopting the implicit regularization as working arena, we discuss in detail how a consistent version of the method can be envisaged, and present many of the subtleties that appear at two-loop order using an abelian chiral model as working example.

ϕ4 theory at seven loops

We give a detailed account of the theory of position space renormalization using graphical functions in the case of dimensionally regularized ${\ensuremath{\phi}}^{4}$ theory in four dimensions. In this theory we calculate the beta function, the mass gamma function, and the self-energy to seven loops in the minimal subtraction scheme. The anomalous dimension $\ensuremath{\gamma}$ is calculated to loop order eight. When possible, we generalize to even dimensions $\ensuremath{\ge}4$ with particular focus on ${\ensuremath{\phi}}^{3}$ theory in six dimensions. In this theory we calculate the anomalous dimension $\ensuremath{\gamma}$ to loop order six.

Electroless Cobalt Deposition on Dealloyed Nanoporous Gold Substrate: A Versatile Technique to Control Morphological and Magnetic Properties.

The connection of multidisciplinary and versatile techniques capable of depositing and modeling thin films in multistep complex fabrication processes offers different perspectives and additional degrees of freedom in the realization of patterned magnetic materials whose peculiar physical properties meet the specific needs of several applications. In this work, a fast and cost-effective dealloying process is combined with a fast, low-cost, scalable electroless deposition technique to realize hybrid magnetic heterostructures. The gold nanoporous surface obtained by the dealloying of an Au40Si20Cu28Ag7Pd5 ribbon is used as a nanostructured substrate for the electrodeposition of cobalt. In the first steps of the deposition, the Co atoms fill the gold pores and arrange themselves into a patterned thin film with harder magnetic properties; then they continue their growth into an upper layer with softer magnetic properties. The structural characterization of the hybrid magnetic heterostructures is performed using an X-ray diffraction technique and energy-dispersive X-ray spectroscopy, while the morphology of the samples as a function of the electrodeposition time is characterized by images taken in top and cross-section view using scanning electron microscopy. Then, the structural and morphologic features are correlated with the room-temperature magnetic properties deduced from an alternating-gradient magnetometer's measurements of the hysteresis loop and first order reversal curves.

Spectral functions of gauge theories with Banks-Zaks fixed points

We investigate spectral functions of matter-gauge theories that are asymptotically free in the ultraviolet and display a Banks-Zaks conformal fixed point in the infrared. Using perturbation theory, Callan-Symanzik resummations, and UV-IR connecting renormalization group trajectories, we analytically determine the gluon, quark, and ghost propagators in the entire complex momentum plane. At weak coupling, we find that a K\"all\'en-Lehmann spectral representation of propagators is achieved for all fields, and determine suitable ranges for gauge-fixing parameters. At strong coupling, a proliferation of complex conjugated branch cuts renders a causal representation impossible. We also derive relations for scaling exponents that determine the presence or absence of propagator nonanalyticities. Further results include spectral functions for all fields up to five loop order, bounds on the conformal window, and an algorithm to find running gauge coupling analytically at higher loops. Implications of our findings and extensions to other theories are discussed.

Tropical Monte Carlo quadrature for Feynman integrals

We introduce a new method to evaluate algebraic integrals over the simplex numerically. This new approach employs techniques from tropical geometry and exceeds the capabilities of existing numerical methods by an order of magnitude. The method can be improved further by exploiting the geometric structure of the underlying integrand. As an illustration of this, we give a specialized integration algorithm for a class of integrands that exhibit the form of a generalized permutahedron. This class includes integrands for scattering amplitudes and parametric Feynman integrals with tame kinematics. A proof-of-concept implementation is provided with which Feynman integrals up to loop order 17 can be evaluated.

Research progress of lock-in amplifiers

The lock-in amplifier can perform high-precision measurement in both time and amplitude dimensions, so that it becomes a key component of instrumental system for precision measurement and control. This article overviews the concept, technology, and application of phase-locked amplifiers as a guide. It first explains the development and evolution of phase-locked amplifiers of analog, digital, and virtual phase-locked amplifiers, demonstrating their relationship and differences. Then, it classifies phase-locked amplifiers from a mathematical perspective based on the order and type of phase-locked loops. Subsequently, the testing process and metrological calibration progress of the main performance of phase-locked amplifiers, such as amplitude, frequency, and phase noise, are introduced. The conversion relationship between key indicators such as phase noise, time-domain jitter, Allan variance, and the coupling relationship with amplitude noise are discussed. Finally, the application forms and effects of phase-locked amplifiers in the fields of spectral enhancement, impedance analysis, magnetic measurement, microscopic imaging, and space exploration are listed. Through some new applications, the prospects of their transition from scientific instruments to industrial and even civilian products through intelligent computing, precise IoT, and other means are briefly given.

Two-loop three-gluon vertex from the Curci-Ferrari model and its leading infrared behavior to all loop orders

We evaluate the three-gluon vertex with one vanishing external momentum within the Curci-Ferrari (CF) model at two-loop order and compare our results to Landau-gauge lattice simulations of the same vertex function for the SU(2) and SU(3) gauge groups in four dimensions. This extends previous works [J. A. Gracey et al., Phys. Rev. D 100, 034023 (2019); N. Barrios et al., Phys. Rev. D 102, 114016 (2020)] that considered similarly the two-loop ghost and gluon two-point functions as well as the two-loop ghost-antighost-gluon vertex (with vanishing gluon momentum). With the parameters of the model being adjusted by fitting the two-point functions to available lattice data, our evaluation of the three-gluon vertex arises as a pure prediction. We find that two-loop corrections systematically improve the agreement between the model and the lattice data as compared to earlier one-loop calculations, with a better agreement in the SU(3) case as already seen in previous studies. We also analyze the renormalization scheme dependence of our calculation. In all cases, this dependence diminishes when two-loop corrections are included, which is consistent with the perturbative CF paradigm. In addition, we study the low momentum regime of the three-gluon vertex in relation with the possibility of zero crossing. Within the CF model, we show that the leading infrared behavior of the exact vertex is given by the same linear logarithm that arises at one-loop order, multiplied by the all orders cubic ghost dressing function at zero momentum (we provide similar exact results for other vertex functions). We argue that this property remains true within the Faddeev-Popov framework under the assumption that the resummed gluon propagator features a decoupling behavior. This shows that the zero crossing is a property of the exact three-gluon vertex function. Within the CF model, we find however that the scale of the zero crossing is considerably reduced when going from one- to two-loop order. This seems consistent with some recent lattice simulations [G. T. R. Catumba et al., EPJ Web Conf. 258, 02008 (2022)]. Finally, our analysis also allows us to support recent claims about the dominance of the tree-level tensor component [F. Pinto-G\'omez et al., arXiv:2208.01020].

Renormalization of supersymmetric chiral theories in rational spacetime dimensions

We renormalize models with scalar chiral superfields with an odd superpotential to several orders in perturbation theory. These extensions of the cubic Wess–Zumino model are renormalizable in spacetime dimensions which are rational. When endowed with an O(N) symmetry it is shown that they share the same property as their non-supersymmetric counterparts in that at a particular fixed point there is an emergent OSp(1|n-1) symmetry, where n is the power of the superpotential. This is shown at a loop order beyond that for which it was established in the parallel non-supersymmetric theory.

Automatic Generation of High-Performance Convolution Kernels on ARM CPUs for Deep Learning

We present <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">FastConv</i> , a template-based code auto-generation open-source library that can automatically generate high-performance deep learning convolution kernels of arbitrary matrices/tensors shapes. FastConv is based on the Winograd algorithm, which is reportedly the highest performing algorithm for the time-consuming layers of convolutional neural networks. ARM CPUs cover a wide range of designs and specifications, from embedded devices to HPC-grade CPUs. The leads to the dilemma of how to consistently optimize Winograd-based convolution solvers for convolution layers of different shapes. FastConv addresses this problem by using templates to auto-generate multiple shapes of tuned kernels variants suitable for skinny tall matrices. As a performance portable library, FastConv transparently searches for the best combination of kernel shapes, cache tiles, scheduling of loop orders, packing strategies, access patterns, and online/offline computations. Auto-tuning is used to search the parameter configuration space for the best performance for a given target architecture and problem size. Results show 1.02x to 1.40x, 1.14x to 2.17x, and 1.22x and 2.48x speedup is achieved over NNPACK, ARM NN, and FeatherCNN on Kunpeng 920. Furthermore, performance portability experiments with various convolution shapes show that FastConv achieves 1.2x to 1.7x speedup and 2x to 22x speedup over NNPACK and ARM NN inference engine using Winograd on Kunpeng 920. CPU performance portability evaluation on VGG–16 show an average speedup over NNPACK of 1.42x, 1.21x, 1.26x, 1.37x, 2.26x, and 11.02x on Kunpeng 920, Snapdragon 835, 855, 888, Apple M1, and AWS Graviton2, respectively.

Non-Fermi liquid behavior and quantum criticality in cubic heavy fermion systems with non-Kramers multipolar local moments

Notable non-Fermi liquid and quantum critical behaviors are observed in rare-earth metallic systems with non-Kramers local moments supporting a number of different multipolar moments. A prominent example is $\text{Pr(Ti,V)}_{2}\text{Al}_{20}$, where the non-Kramers doublet of the $\text{Pr}^{3+}$ ion allows quadrupolar and octupolar moments, but lacks a dipolar moment. Previous theoretical studies show that a single impurity Kondo problem with such an unusual local moment leads to novel non-Fermi liquid states. In this work, we investigate possible quantum critical behaviors arising from the competition between non-Fermi liquid states and multipolar-ordered phases induced by the RKKY interaction. We consider a local version of the corresponding Kondo lattice model, namely the Bose-Fermi Kondo model. Here, the multipolar local moments are coupled to fermionic and bosonic bath degrees of freedom representing the multipolar Kondo effect and RKKY interactions. Using a perturbative renormalization group (RG) study up to two loop order, we find critical points between non-Fermi liquid Kondo fixed points and a quadrupolar ordered fixed point. The critical points describe quantum critical behaviors at the corresponding phase transitions and can be distinguished by higher order corrections in the octupolar susceptibility that can be measured by ultrasound experiments. Our results imply the existence of a rich expansion of the phases and quantum critical behaviors in multipolar heavy fermion systems.

QED as a many-body theory of worldlines: General formalism and infrared structure

We discuss a reformulation of QED in which matter and gauge fields are integrated out explicitly, resulting in a many-body Lorentz covariant theory of 0+1 dimensional worldlines describing super-pairs of spinning charges interacting through Lorentz forces. This provides a powerful, string inspired definition of amplitudes to all loop orders. In particular, one obtains a more general formulation of Wilson loops and lines, with exponentiated dynamical fields and spin precession contributions, and worldline contour averages exactly defined through first quantized path integrals. We discuss in detail the attractive features of this formalism for high order perturbative computations. We show that worldline S-matrix elements, to all loop orders in perturbation theory, can be constructed to be manifestly free of soft singularities, with infrared (IR) divergences captured and removed by endpoint photon exchanges at infinity that are equivalent to the soft coherent dressings of the Dyson S-matrix proposed by Faddeev and Kulish. We discuss these IR structures and make connections with soft theorems, the Abelian exponentiation of IR divergences and cusp anomalous dimensions. Follow-up papers will discuss the efficient computation of cusp anomalous dimensions and universal features of soft theorems.

Squeezing as a probe of the universality hypothesis

We compute analytically the radiative quantum corrections, up to next-to-leading loop order, to the universal critical exponents for both massless and massive O(N) [Formula: see text] scalar squeezed field theories for probing the universality hypothesis. For that, we employ six distinct and independent methods. The outcomes for the universal squeezed critical exponents obtained through these methods are identical among them and reduce to the conventional ones where squeezing is absent. Although the squeezing mechanism modifies the internal properties of the field, the squeezed critical indices are not affected by the squeezing effect, thus implying the validity of the universality hypothesis, at least at the loop level considered. At the end, we present the corresponding physical interpretation for the results in terms of the geometric symmetry properties of the squeezed field.

Dephasing effects on the low-energy dynamics of ϕ4-model

In this paper, we investigate the Lindblad dynamics arising due to single particle loss and local dephasing in a d-dimensional nonrelativistic quantum system. Using Schwinger–Keldysh functional techniques, we derive the static mean field equation for the symmetry broken phase. Fluctuations over mean field are studied within the Bogoliubov approximation, and the momentum distribution and effective temperature are calculated. An effective temperature arises which depends on the dephasing rate [Formula: see text] and mean field particle density. Furthermore, a perturbative renormalization group (RG) upto one loop order is used to study the underlying dynamics of the model. The conditions for the system to be perturbative are obtained.

Nonextensive percolation and Lee-Yang edge singularity from nonextensive λϕ3 scalar field theory

We compute the critical exponents for nonextensive λϕ3 scalar field theory for all loop orders and |q−1|<1. We apply the results for both nonextensive percolation and Lee-Yang edge singularity problems. The corresponding systems are nonextensive generalizations of their extensive counterparts. For that we employ tools from the recently introduced nonextensive statistical field theory. The results for the nonextensive critical exponents computed depend on the nonextensive parameter q, which encodes global correlations among the degrees of freedom of the system. The extensive results are recovered in the limit q→1. There is an interplay between global correlations and fluctuations, once the nonextensive critical exponents depend on q. This dependence is in agreement with the universality hypothesis.

Stampedes I: fishnet OPE and octagon Bootstrap with nonzero bridges

Some quantities in quantum field theory are dominated by so-called leading logs and can be re-summed to all loop orders. In this work we introduce a notion of stampede which is a simple time-evolution of a bunch of particles which start their life in a corner — on the very right say — and hop their way to the opposite corner — on the left — through the repeated action of a quantum Hamiltonian. Such stampedes govern leading logs quantities in certain quantum field theories. The leading euclidean OPE limit of correlation functions in the fishnet theory and null double-scaling limits of correlators in mathcal{N} = 4 SYM are notable examples. As an application, we use these results to extend the beautiful bootstrap program of Coronado [1] to all octagons functions with arbitrary diagonal bridge length.

Pentagon Wilson loop with Lagrangian insertion at two loops in mathcal{N} = 4 super Yang-Mills theory

We compute the two-loop result for the null pentagonal Wilson loop with a Lagrangian insertion (normalized by the Wilson loop without insertion) in planar, maximally supersymmetric Yang-Mills theory. This finite observable is closely related to the Amplituhedron, and it is reminiscent of finite parts of planar two-loop five-particle scattering amplitudes. We verify that, up to this loop order, the leading singularities are given by the same conformally invariant expressions that appear in all-plus pure Yang-Mills amplitudes. The accompanying weight-four transcendental functions are expressed in terms of the pentagon functions space known from planar two-loop five-particle amplitudes, but interestingly only a subset of the functions appears. Being a function of four dimensionless variables, the observable has interesting asymptotic limits. We verify that our analytic result is consistent with soft and collinear limits, and find an intriguingly simple pattern in the multi-Regge limit. Thanks to the new result we can also conjecturally predict, for general kinematics, the maximal weight piece of the planar three-loop five-particle all-plus amplitude in pure Yang-Mills theory. Motivated by the Amplituhedron geometry, we investigate positivity properties of the integrated answer. Generalizing previous results at four particles, we find numerical evidence that the two-loop five-particle result has uniform sign in a kinematic region suggested by the loop Amplituhedron.